Convergent Analytic Solutions for Homoclinic Orbits in Reversible and Non-reversible Systems

In this paper, convergent, multi-infinite, series solutions are derived for the homoclinic orbits of a canonical fourth-order ODE system, in both reversible and non-reversible cases. This ODE includes traveling-wave reductions of many important nonlinear PDEs or PDE systems, for which these analytical solutions would correspond to regular or localized pulses of the PDE. As such, the homoclinic solutions derived here are clearly topical, and they are shown to match closely to earlier results obtained by homoclinic numerical shooting. In addition, the results for the non-reversible case go beyond those that have been typically considered in analyses conducted within bifurcation-theoretic settings. We also comment on generalizing the treatment here to parameter regimes where solutions homoclinic to exponentially small periodic orbits are known to exist, as well as another possible extension placing the solutions derived here within the framework of a comprehensive categorization of ALL possible traveling-wave solutions, both smooth and non-smooth, for our governing ODE.Comment: arXiv admin note: text overlap with arXiv:math-ph/060606

Extensions of the General Solution to the Inverse Problem of the Calculus of Variations, and Variational, Perturbative and Reversible Systems Approaches to Regular and Embedded Solitary Waves

In the first part of this Dissertation, hierarchies of Lagrangians of degree two, three or four, each only partly determined by the choice of leading terms and with some coefficients remaining free, are derived. These have significantly greater freedom than the most general differential geometric criterion currently known for the existence of a Lagrangian and variational formulation since our existence conditions are for individual coefficients in the Lagrangian. For different choices of leading coefficients, the resulting variational equations could also represent traveling waves of various nonlinear evolution equations. Families of regular and embedded solitary waves are derived for some of these generalized variational ODEs in appropriate parameter regimes. In the second part, an earlier approach based on soliton perturbation theory is significantly generalized to obtain an analytical formula for the tail amplitudes of nonlocal solitary waves of a perturbed generalized fifth-order Korteweg-de Vries (FKdV) equation. On isolated curves in the parameter space, these tail amplitudes vanish, producing families of localized embedded solitons in large regions of the space. Off these curves, the tail amplitudes of the nonlocal waves are shown to be exponentially small in the small wavespeed limit. These seas of delocalized solitary waves are shown to be entirely distinct from those derived in that earlier work. These perturbative results are also discussed within the framework of known reversible systems results for various families of homoclinic orbits of the corresponding traveling-wave ordinary differential equation of our generalized FKdV equation. The third part considers a variety of dynamical behaviors in a multiparameter nonlinear Mathieu equation with distributed delay. A slow flow is derived using the method of averaging, and the predictions from that are then tested against direct numerical simulations of the nonlinear Mathieu system. Both areas of agreement and disagreement between the averaged and full numerical solutions are considered

Solitary wave families of NLPDES via reversible systems theory

The Ostrovsky equation is an important canonical model for the undirectional propagation of weakly nonlinear long surface and internal waves in a rotating, mviscid and incompressible fluid. Since solitary wave solutions often play a central role in the long-time evolution of an inital disturbance. we consider such solutions here (via the normal form approach) within the framework of reversible system theory. Resides confirming the existence of the known family of solitary waves and its reduction to the Kdv limit. w we find a second family of multihumped (or N-pulse) solutions, as well as a contimum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solutions. The second and third families of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new directions for future work, including on other NLPDEs, are also mentioned. (C) 2009 IMACS Published by Elsevier B.V. All rights reserved

Solitary Wave Families Of Nlpdes Via Reversible Systems Theory

The Ostrovsky equation is an important canonical model for the unidirectional propagation of weakly nonlinear long surface and internal waves in a rotating, inviscid and incompressible fluid. Since solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves and its reduction to the KdV limit, we find a second family of multihumped (or N-pulse) solutions, as well as a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The second and third families of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work, including on other NLPDEs, are also mentioned. © 2009 IMACS